Question

Suppose that the S&P 500, with a beta of 1.0, has an expected return of 10% and Treasury bills provide a risk-free return of 4%.

a. How would you construct a portfolio from these two assets with an expected return of 8%? Specifically, what will be the weights in the S&P 500 versus T-bills

b. How would you construct a portfolio from these two assets with a beta of 0.4?

c. Find the risk premiums of the portfolios in (a) and (b), and show that they are proportional to their betas.

Answer 1

1.

Let weight of S&P500 be w and weight of Treasury bills be 1-w

So, expected return=w*10%+(1-w)*4%=4%+6%*w

=>8%=4%+6%*w

=>w=0.67

So, weight of S&P500=0.67 or 67% and weight of Treasury bills=1-0.67=0.33 or 33%

Beta of portfolio=0.67*1=0.67

Risk premium=8%-4%=4%

2.

Let weight of S&P500 be w and weight of Treasury bills 1-w

As beta of risk free or Treasury bills is zero

Hence, beta of portfolio=w*1+(1-w)*0=w

=>0.4=w

Hence, weight of S&P500=0.4 or 40% and weight of Treasury Bills=1-0.4=0.6 or 60%

Expected return=0.4*10%+0.6*4%=6.4%

Risk premium=6.4%-4%=2.4%

3.

Risk premium for or a)=4%

Risk premium for b)=2.4%

Risk premium a)/b)=4/2.4=1.67

Beta a)/b)=0.67/0.4=1.67

Hence risk premium is proportional to beta